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1
Determining Molecular Structure:
An Experimental Simulation of X-ray Diffraction
I. Introduction to Diffraction
How do we know how atoms arrange themselves in molecules and extended solids?
Information regarding bond distances and angles, the basis for the familiar structures we draw on
the board in class, can be obtained by detailed analysis of X-ray, neutron, or electron diffraction
data. In this lab, we will simulate an X-ray diffraction experiment to begin to understand how
the analysis of diffraction data allows us to obtain precise information about the structures of
molecules and extended solids.
Diffraction of a wave by a periodic array is due to phase differences that result in
constructive and destructive interference. Diffraction can occur when waves pass through a
periodic array if the repeat distance of the array is similar to the wavelength of the waves. In
crystalline solids, the atoms are arranged in repeating three-dimensional arrays or lattices with
repeating distances of about 10-10 m. The observation of diffraction patterns when beams of
electrons, neutrons or X-rays interact with crystalline solids provides evidence for the wave
nature of those beams and for the periodic arrangement of atoms in the crystalline solids. Over
the past seventy-five years, X-ray diffraction has developed into a powerful and convenient
technique used by chemists to determine the structures of newly prepared molecules and solids
and even to unravel the structures of complex proteins.
In a typical single crystal X-ray diffraction experiment, a beam of monochromatic X-rays
is collimated onto a crystal (Figure 1.1). Diffraction by the periodic array of atoms in the crystal
leads to a complex pattern of constructive and destructive interference at the detection plane, in
this case, a photographic plate. The symmetry and spacings of the periodic arrangement of atoms
in the crystal are reflected in the symmetry and spacings of the resulting diffraction pattern. The
details of the diffraction pattern, when mathematically analyzed, can provide the details of the
three-dimensional arrangement of atoms in the crystal itself.
Rather than do a single crystal X-ray diffraction experiment in this lab, we are going to
make a change of scale and study the diffraction of visible light. By using periodic arrangements
of dots with repeating spacings of about 10-4 m (produced via a laser printer and photographic
reduction) instead of atoms with spacings of about 10-10 m, visible light can be used instead of Xray light to create diffraction patterns. In this experiment, you will shine red laser light ( = 633
nm) or green laser light ( = 544 nm) through a 35 mm slide containing repeating patterns of
dots, and observe a type of diffraction known as Fraunhofer diffraction.
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X-Ray Diffraction
X-ray
Tube
High
Voltage
Crystal
X-ray
Beam
Lead Screen
Photographic
Plate
Figure 1.1. A comparison of the apparatus used for X-ray diffraction and the optical transform
experiment.
Bragg diffraction, named after the diffraction pioneer Sir William Bragg, is the type of
diffraction that is observed in most X-ray diffraction experiments. Fraunhofer and Bragg
diffraction are compared in Figure 1.2. In both types of diffraction, the path difference between
any two waves must be a whole number of wavelengths if the waves are to arrive at a detector in
phase and result in constructive interference. A simple equation describing the condition for
constructive interference can be derived from this path difference and trigonometry.
Mathematically, the equations for Fraunhofer and Bragg diffraction differ only by a factor of two
and embody the same functional dependence on the repeating distance, d, the wavelength, , and
the scattering angle ( or ).
In this experiment, you will first check how the size and orientation of the diffraction
pattern is related to the periodic array that produced it, and then you will measure distances in the
diffraction pattern in order to calculate the repeat distance for the array on the slide. By
measuring the distances X and L, shown in Figure 1.1, and using the trigonometry definition that
tan  = X/L, you can solve for . Use of the Fraunhofer equation, d sin  = n then gives d when
 is known.
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Figure 1.2. A comparison of Fraunhofer diffraction with Bragg diffraction. Note the conditions
for constructive interference are mathematically similar, differing only by a factor of two.
II. Introduction to Symmetries of the Plane
The periodic patterns that we will consider in this experiment are characterized by
translational symmetry in two dimensions. As a result, the essential characteristics of the
repeating pattern can be captured in a unit cell, a parallelogram that can be translated in
directions parallel to its edges to reproduce the pattern. In addition to translational symmetry, the
periodic patterns may also possess additional symmetry elements such as rotational symmetry,
mirror symmetry, or glide symmetry. Rotational symmetry is described by invariance to rotations
of 360°/n around an imaginary axis perpendicular to the plane (an n-fold rotation). Mirror
symmetry relates two portions of the pattern by reflection across an imaginary mirror plane.
Glide symmetry, the symmetry of footprints on a beach, is comprised of reflection across a
mirror plane followed by translation parallel to the mirror plane. The complete collection of all
of the symmetry operations present in a repeating pattern determines the plane group of that
pattern.
The possible symmetries of two-dimensional repeating patterns are described by the 17
plane groups. Each plane group is named with a symbol such as p1, cmm, or p4gm. Examples of
periodic patterns belonging to each of the 17 plane groups are shown in Figure 1.3. All of these
patterns were generated by applying the symmetry of the group to the same triangular shape or
asymmetric unit. The result is 17 very different periodic arrays. While the possibilities for
unique patterns seem infinite, choices are limited by the rules of symmetry and mathematics. For
example, when rotational symmetry is combined with translational symmetry, only certain types
of rotations are permitted. Pentagonal or 5-fold rotational symmetry is forbidden in patterns with
translational symmetry. This can be proven mathematically, but a conceptual way to understand
this is to remember that pentagons cannot be used to infinitely tile space in two dimensions.
Rectangles, squares, triangles, and hexagons, on the other hand, can all be used to tile a plane.
Thus, 2-fold, 3-fold, 4-fold, and 6-fold rotational symmetry axes are all allowed components of
two-dimensional periodic patterns. The presence of higher order rotation axes (n > 2) can be
deduced from the plane group symbol (i.e. p4, p31m, and p6mm)
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Removed for copyright reasons.
Please see:
I. Hargittai and G. Lengyel, “The Seventeen Two-Dimensional Space-Groups in
Hungarian Needlework,” J. Chem. Educ. 1985, 62, 35-36.
Figure 1.3. Examples of patterns from each of the 17 plane groups of symmetry. Images adapted
from I. Hargittai and G. Lengyel, “The Seventeen Two-Dimensional Space-Groups in Hungarian
Needlework,” J. Chem. Educ. 1985, 62, 35-36.
Assigning a particular repeating pattern to one of the 17 plane groups involves identifying
the presence of key symmetry elements and their locations with respect to each other. Figure 1.4
shows the locations of all the symmetry elements in each of the 17 plane groups with respect to a
standard choice of unit cell.1 For example, patterns belonging to p1 only exhibit translational
symmetry; p2, p3, p4, and p6 are pure rotational groups with no mirror or glide symmetry. Both
p31m and p3m1 contain 3-fold rotation axes, mirror planes, and glide planes, however the key to
distinguishing them lies in the locations of the 3-fold axes. In p3m1, all the 3-fold rotation axes
are located at the intersections of mirror planes, whereas in p31m, one set of 3-fold rotation axes
occupies unique locations within the unit cell. With some practice, this systematic identification
of symmetry elements can be used to easily categorize the plane group of any periodic two1
Images were adapted from the International Tables of Crystallography, Volume I, and follow the conventions and
notation of the International Union of Crystallography.
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dimensional pattern. The following website offers an excellent table summarizing the
distinguishing characteristics and many examples of tilings belonging to each of the 17 plane
symmetry groups: http://www2.spsu.edu/math/tile/symm/ident17.htm
p1
pm
pg
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2
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pmg
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3
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p
31
p
p
m
4
p
m
6
3
p
p
6
m
4
1
g
m
Figure 1.4. Unit cells and locations of the symmetry elements for each of the 17 plane groups.
Shortened group names are used here in comparison to the names used in Figure 1.3. Perpendicular
rotation axes intersect the plane at the locations indicated by lenses (2-fold), triangles (3-fold),
squares (4-fold), or hexagons (6-fold). Mirror planes are shown as solid lines (not to be confused
with the edges of the unit cell in some cases). Glide planes are indicated by dashed lines.
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III. Experimental
The Slide Patterns
The periodic patterns that we will be using as our “crystals” for this diffraction
experiment are shown on the next page. In this case, we already know what the periodic patterns
look like. Our interest is to examine the diffraction patterns caused by these “crystals” and to
observe the effects that changing the symmetries and spacings in the “real pattern” have on the
diffraction patterns. A crystallographer, on the other hand, is usually working backwards from
the diffraction pattern to try and construct the periodic pattern causing the diffraction. The
structure of the crystal is initially unknown, but the diffraction pattern offers clues to the
symmetry of the underlying periodic pattern. Each plane group usually leaves one or more
characteristic “calling cards” in the diffraction pattern. We will try to discover the nature of
some of these signatures through the course of this experiment.
• Unit Cells and Plane Groups for the Slide Patterns
Before you come to lab…
You should begin by identifying the plane groups for each of the eight periodic patterns (a thru h)
on the next page. Assume that these patterns continue infinitely in the plane, i.e., do not worry
about edges of the pattern. Also, with the exception of pattern "g," assume that the squares that
make up the pattern are actually atoms with spherical symmetry.
In your lab notebook...
Note in the remainder of this handout, questions that you are to address in your lab
notebook will be written in Arial font.
1) Identify the plane group for each slide pattern. Two or more patterns may belong to the same
plane group.
2) Draw a unit cell for each of the eight patterns with "atoms" (squares) centered on the corners that
matches the shape and orientation of the conventional unit cell for the plane group as shown in
Figure 1.4. Note that Pattern "g" will not have a unit cell with “atoms” on the corners.
3) Identify the symmetry elements (mirror planes, rotational axes, and glide planes) present in each
of the eight patterns by drawing in their locations on your unit cells. These should be symmetry
elements of the infinitely repeating pattern. Different color pens are very useful here! Refer to
Figure 1.4 for guidance.
Once you arrive in lab…
Obtain a 35-mm slide containing the slide patterns. The photographic slide contains eight
patches. Each patch has a different periodic pattern of dots (actually little squares). The
repeating patterns in each of the eight patches are shown on the next page. You can identify
which patch of the slide corresponds to which pattern by orienting the slide so that two dark
patches are in the upper left corner. The patterns on the paper are labeled "a", "b", "c", etc. For
ease of interpretation, you should keep the slide and the pattern sheet oriented the same way.
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The Diffraction Patterns
• Diffraction of a White Light Source
Find a point source of white light and look at it through the slide.
In your lab notebook...
Describe what you observe in your lab notebook, being explicit about any wavelength dependence
you observe. Referring back to Figure 1.2, explain how your observations support the wavelength
dependence in the Fraunhofer diffraction equation. For different wavelengths, which parameter do
you observe changing?
• Diffraction of a Monochromatic Source
Several lasers are set up in the laboratory (red lasers,  = 633 nm, and green laser,  = 544
nm) to observe diffraction patterns.
CAUTION:
DO NOT LOOK DIRECTLY INTO THE LASER BEAM!
BE CAREFUL NOT TO SHINE THE LASER TOWARDS OTHER PEOPLE!
You should look only at diffuse reflections of the beam from rough surfaces such as
paper or the wall.
To get started, hold the slide in the laser beam and observe the diffraction patterns. Practice
shining the laser through each of the eight patches on the slide. It is easier to move the slide
than the laser. For serious measurements of spot positions, you will need to hold the laser
and slide stationary. Appropriate clamps are available in the laboratory.
In your lab notebook...
• Record the color and wavelength of your laser.
• Measure and record the distance from the slide to the surface on which you project the
diffraction pattern.
For each dot pattern, your data will consist of a detailed record of the resulting diffraction
pattern. Draw pictures of the diffraction patterns in your notebook, drawing several rows of
spots in each direction around the bright central spot (as in Figure 1.5). Record a relative
measure of intensity by drawing larger circles for more intense spots and smaller circles for
spots of weaker intensity. Using a ruler, measure and record relevant distances between
spots, and estimate the uncertainties of your measurements.
Work through the patterns in the order described below, first drawing the
diffraction patterns and then answering any questions directly in your
notebook.
Answer the questions with complete sentences so that a reader of your lab
notebook does not require this write-up to understand your observations.
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(-1,1)
(0,1)
(1,1)
(2,1)
(-1,0)
(0,0)
(1,0)
(2,0)
(-1,-1)
(0,-1)
(1,-1)
(2,-1)
(-1,-2)
(0,-2)
(1,-2)
(2,-2)
Figure 1.5. Typical diffraction pattern observed from a simple square lattice. Diffraction spots
are indexed with a set of integers, (h, k), to indicate the order of diffraction in a particular
direction.
1) Begin with patterns "b" and "d". Both are simple square lattices. Pattern "d" has a lattice
spacing that is 1.5 times the lattice spacing of pattern "b". Switch back and forth between "b"
and "d".
• Draw each diffraction pattern in your lab notebook.
• Do these diffraction patterns have 2-fold, 3-fold, 4-fold, or 6-fold rotational symmetry? How does
the symmetry of the diffraction pattern compare to the symmetry of the dot pattern on the slide?
• Does the smaller or larger lattice give the smaller diffraction pattern?
• Measure and record the horizontal and vertical spacings in the diffraction patterns of "b" and "d".
2) The slide pattern "c" has the same repeat distance as slide pattern "d", but a larger atom is
positioned in the center of each unit cell in pattern "c".
• Draw the diffraction pattern in your lab notebook.
• Compare diffraction patterns of "c" and "d". It is useful to quickly move the slide back and forth
between patterns "c" and "d" and observe what happens to the intensities of the spots.
Hint: Look for a subset of spots that “blinks” brighter when you go from one pattern to the next.
• Identify the indices, (h, k), of the subset of spots that grows brighter in moving from one pattern
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to the next. Do you note any patterns in these indices (as in even or odd numbers)?
• Measure and record the horizontal and vertical spacings in the diffraction pattern of "c".
3) Now compare patterns "a" and "b". In the slide pattern "a", the atom in the center of the unit
cell is exactly the same size as the atoms on the corners of the unit cell. This is called a
centered unit cell. (Note that the unit cell in "c" is strictly not a centered unit cell since the
atom in the center is not identical to the atoms on the corners.) Shift the slide back and forth
between "a" and "b" and notice the effect of the centered cell on the diffraction pattern.
The presence of a centered unit cell causes a set of systematic absences in a diffraction
pattern, that is, a pattern of missing spots. An alternative view is that the extra symmetry of
centering sets up an additional condition for constructive interference. The extra condition
limiting possible reflections for a centered unit cell is that h + k = 2n. In other words, the
sum of the h and k indices of a given diffraction spot must be an even integer.
• Draw the diffraction pattern for "a" in your lab notebook.
• Examine the evidence that the condition of systematic absences is obeyed in the diffraction
pattern of "a". Assuming that Figure 1.5 represents the diffraction pattern for "b", what subset of
spots disappears for "a"? What pattern is there in the (h, k) indices of the disappearing spots?
In addition to the centered unit cell, another unit cell, called a primitive unit cell can be drawn
for slide pattern "a" that only has atoms on the corners.
• Draw both a primitive and a centered unit cell for slide pattern "a" showing the relative orientations
of these unit cells in the drawings on page 7.
• Measure and record the horizontal, vertical, and diagonal spacings in the diffraction pattern of
"a" that will help you determine the sizes of the two unit cells you have drawn above.
4) Patterns "e" (rectangular symmetry) and "f" (parallelogram) have less than square symmetry.
• Observe and draw their diffraction patterns.
• Measure the spacings in the diffraction pattern of "e". Why does it appear as if both of these
diffraction patterns have been rotated from what they should be? Beware! Appearances here
are deceiving!
5) Pattern "g" has glide symmetry. The unusual content of the unit cell has unusual effects on
the intensity pattern.
• Observe and draw the diffraction pattern of "g".
The presence of the glide plane leaves a signature in the diffraction pattern that a
crystallographer knows how to interpret. If the glide plane is oriented vertically in the
periodic pattern, the central row of vertical spots in the diffraction pattern, spots indexed
(0,k), will obey the additional condition that k = 2n. If the glide plane is oriented horizontally
in the periodic pattern, the central horizontal row of spots in the diffraction pattern will be
affected: (h, 0) h = 2n.
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• Describe the orientations of the glide planes in the periodic pattern of "g" and discuss how the
diffraction evidence supports the presence of this glide symmetry.
6) Pattern "h" has hexagonal symmetry, as does its diffraction pattern.
• Observe and draw the diffraction pattern of "h".
• Hexagonal symmetry means the presence of 6-fold rotational symmetry. Discuss the diffraction
evidence for hexagonal symmetry with appropriate measurements from the diffraction pattern.
III. Analysis and Conclusions
Calculate the lattice spacing for patterns "a" through "e" from the diffraction equation.
Note: The angle  is sufficiently small that sin can be replaced by  (measured in radians);
recall that 2π radians equals 360°, so 1 radian = 57.3°. The equation then reduces to n  = d .
The wavelength of the laser is 633 nm or 544 nm, and n = 1 for the first order (smallest angle)
diffraction spot.
• Show a sample calculation for the sizes of the two possible unit cells you drew for "a" in step (3)
above. Indicate clearly the unit cell repeat distances for the primitive and the centered cells.
Remember that the orientation of the distances you measure in the diffraction pattern matches
the orientation of distances in the dot pattern.
• Report your results for patterns "a" through "e" in a table containing a sketch of the unit cell and
unit cell parameters in mm for each pattern. Remember that you should have 2 distances
calculated for pattern "e."
Because of the way the patterns were produced (originating as pixels on a computer screen), the
spacings for patterns "a" through "e" should all be whole number multiples of the pixel size.
• What is your best estimate for the size of one dot (pixel) on the slide? Hint: Do not forget that
pattern "a" resulted in unusual systematic absences in the diffraction pattern. It may be useful
to compare the sizes of the unit cells you drew for patterns "a" and "b" and make sure that your
calculated lattice spacings agree.
• Finally, what general conclusions can you make about the relationships of the symmetry and
spacings of the dot patterns to the symmetries and spacings of the diffraction patterns?
Page 11